Discrete breathers in honeycomb Fermi–Pasta–Ulam lattices

Abstract

We consider the two-dimensional Fermi-Pasta-Ulam lattice with hexagonal honeycomb symmetry, which is a Hamiltonian system describing the evolution of a scalar-valued quantity subject to nearest neighbour interactions. Using multiple-scale analysis we reduce the governing lattice equations to a nonlinear Schrödinger (NLS) equation coupled to a second equation for an accompanying slow mode. Two cases in which the latter equation can be solved and so the system decoupled are considered in more detail: firstly, in the case of a symmetric potential, we derive the form of moving breathers. We find an ellipticity criterion for the wavenumbers of the carrier wave, together with asymptotic estimates for the breather energy. The minimum energy threshold depends on the wavenumber of the breather. We find that this threshold is locally maximised by stationary breathers. Secondly, for an asymmetric potential we find stationary breathers, which, even with a quadratic nonlinearity generate no second harmonic component in the breather. Plots of all our findings show clear hexagonal symmetry as we would expect from our lattice structure. Finally, we compare the properties of stationary breathers in the square, triangular and honeycomb lattices.

Asymptotic analysis of breather modes in a two-dimensional mechanical lattice

“…and B given by (A.17). This is the same form of equation as obtained for the scalar two-dimensional lattices analysed previously [8,9,33]. To obtain solutions which are localised in both spatial dimensions, we require that the spatial derivatives are elliptic in nature.…”

“…We have previously considered two-dimensional electrical transmission lattices composed of inductors and nonlinear capacitors, in which there is only a scalar unknown, the charge Q m,n (t), at each lattice site (m, n) [31], covering the square [8], triangular [9] and honeycomb arrangements [33]. These cases are considerably simpler than the mechanical system considered by Marin et al [23,24], in which a two-component vector has to be found at each node, and in such systems, the unknown horizontal and vertical displacements are intrinsically coupled together.…”

Bright and Dark Soliton Pulse in Solid Core Photonic Crystal Fibers

“…If λ > 0, the solution for soliton is known as bright soliton, and if λ < 0, the solution is known as dark soliton (Chen et al, 2012;Wattis & James, 2014).…”

“…The nonlinear processes in photonic crystal fibers (PCFs) that generate soliton production are typically weak and Kerr effect, resulting in a local index change that is exactly proportional to the amplitude (Kartashov et al, 2011). The primary nonlinear equation controlling the pulse progression in this instance is the widely recognized nonlinear Schroeder (NLS) equation for the challenging and intricate electric field magnitude packet (Wattis & James, 2014). This equation features two different types of localized options: bright and dark solitons, which vary based on the group-velocity dispersion's (GVD) magnitude (Talla Mbé et al, 2017).…”

Wave propagation and pattern formation in two-dimensional hexagonally-packed granular crystals under various configurations

“…Aside from two-dimensional granular crystals composed of square lattice packings, there are a variety of other geometric configurations which have been studied. Wattis et al [17] investigated wave propagation on a honeycomb lattice within the context of a model is based on Kirchoff's laws of electrical charge on each node. The dispersion relation is shown to have two branches, acoustic and optical, and they will meet at the Dirac point.…”